# coends of stable infinity categories

Let $\mathcal{I}$ be a small ordinary category that I would like to think of as a diagram category. (If it helps: In my application $\mathcal{I}$ has only one object, i.e. comes from a monoid). Denote the $\infty$-category of small stable $\infty$-categories and exact functors by $Cat^{\rm ex}_{\infty}$. Suppose I have given a functor $$F \colon \mathcal{I} \times \mathcal{I}^{\rm op} \to Cat^{\rm ex}_{\infty}$$ where I am secretly regarding the category on the left as an $\infty$-category.

Are coends defined for $\infty$-functors as above? Does the coend of $F$ as above always exist?

If the answer to the above questions is "yes": Can we describe the weak Kan complex underlying the coend in some way?

• Interesting question. I was led to something similar when I asked how are weighted limits defined in this setting. But why are you limiting to stable $\infty$-categories? Mar 4, 2014 at 20:05

There is some discussion of coends in $\infty$-categories in my PhD thesis (Sheffield, 2010). I think that, given that the target category $\mathrm{Cat}_\infty^\mathrm{ex}$ is cocomplete, this gives you one way to do it.